Relations between exceptional sets for additive problems

TL;DR

This paper introduces a method to bound the set of integers not represented by additive forms, using subforms, with applications to Waring's problem for cubes, showing the count of exceptions grows as O(N^{3/7}).

Contribution

It presents a novel approach to relate exceptional sets of additive forms to those of subforms, providing new bounds in Waring's problem for cubes.

Findings

Bound on non-representable integers as O(N^{3/7})

Method relates exceptional sets of forms to subforms

Applications to Waring's problem for cubes

Abstract

We describe a method for bounding the set of exceptional integers not represented by a given additive form in terms of the exceptional set corresponding to a subform. Illustrating our ideas with examples stemming from Waring's problem for cubes, we show, in particular, that the number of positive integers not exceeding N, that fail to have a representation as the sum of six cubes of natural numbers, is O(N^{3/7}).